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Number Theory/Joint Proportionality



In one of my past question I asked how to solve the following:

4 carpenters can build 8 houses in 10 days.  2 carpenters can build how many houses in 15 days?

Answer: 6 houses

You provided this solution:

"we can use the idea of joint proportionality.  The number of houses built is jointly proportional to the number of carpenters and the number of days.  So h = kcd where k is a constant.  Substituting the given: 8 = k*40, so k = 1/5.  Our formula is then h = cd/5."

How was the idea of joint proportionality determined?  For example, how did you know to use h = kcd and not d = khc or c = khd?

I thank you for your reply.

ANSWER: Hello Kenneth
If we double the number of carpenters, then we double the number of houses, so h is proportional to c.
If we double the number of days, we double the number of houses, so h is proportional to d.  So h is proportional to both c and d.  So it is proportional to the product.  This is the principle of joint variation.  Then h = kcd.
Hope that clarifies the problem.
Best wishes

---------- FOLLOW-UP ----------


I want to thank you for your reply.

I understand the calculation but not for all situations.  Here is another example:

24 men working 8 hours per day can do a piece of work in 15 days  In how many days can 20 men working at 9 hours per day do the same work?

If men are increased the time, hours and days, will decrease.  Time and days are indirectly proportional to men.  How is the solution determined for this example?

I thank you for your help and assistance.

Hello Kenneth

We are trying to find a number of days, so it is best to have d as the subject.
d is inversely (not indirectly) proportional to both men,m, and hours, h. So d = k/(mh).  Substituting the given gets us to k = 2880, and then using the formula gives the answer 16 days.

Best wishes

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